Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

In problems 1-6, determine the convergence set of the given power series.
$\sum_{n = 0}^{\infty} \frac{(n + 2)!}{n!} {(x + 2)}^{n}$

The set is, $x \in (- 3, - 1)$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:To Find the Radius of convergence

Use the ratio test to determine the radius of convergence (with ): $a_{n} = \frac{(n + 2)!}{n!}$

$\begin{matrix} \lim_{n \to \infty} |\frac{a_{n}}{a_{n + 1}}| = \lim_{n \to \infty} |\frac{\frac{(n - 2)!}{n!}}{\frac{(n + 3)!}{(n + 1)!}}| \\ = \lim_{n \to \infty} |\frac{(n + 2)!}{(n + 3)!} \cdot \frac{(n + 1)!}{n!}| \\ = \lim_{n \to \infty} |\frac{(n + 2)!}{(n + 3) (n + 2)!} \times \frac{(n + 1) n!}{n!}| \\ = \lim_{n \to \infty} |\frac{(n + 1)}{n + 3}| \\ = \lim_{n \to \infty} |\frac{(1 + (1 / n))}{1 + (3 / n)}| \\ = 1 \end{matrix}$

The radius of convergence is 1, therefore convergent set for the given power series is. $| x + 2 | < 1$

Step 2: Find the set of convergence

To completely identify the convergence set, we have to check whether the boundary points -1 and -3 are included in the set or not.

Checking at $x = - 1$ , by substituting the value ofx by -1 .

$\begin{matrix} \sum_{n = 0}^{\infty} \frac{(n + 2)!}{n!} {(x + 2)}^{n} = \sum_{n = 0}^{\infty} \frac{(n + 2)!}{n!} {(- 1 + 2)}^{n} \\ = \sum_{n = 0}^{\infty} \frac{(n + 2)!}{n!} {(1)}^{n} \\ = \sum_{n = 0}^{\infty} \frac{(n + 2) (n + 1) n!}{n!} \\ = \sum_{n = 0}^{\infty} (n + 2) (n + 1) \\ = \infty \end{matrix}$

The above series is divergent, thus the point -1 is excluded from the convergent set

Similarly, checking at $x = - 3$ by substituting the value of x by -3.

$\begin{matrix} \sum_{n = 0}^{\infty} \frac{(n + 2)!}{n!} {(x + 2)}^{n} = \sum_{n = 0}^{\infty} \frac{(n + 2)!}{n!} {(- 3 + 2)}^{n} \\ = \sum_{n = 0}^{\infty} \frac{(n + 2)!}{n!} \\ = \sum_{n = 0}^{\infty} \frac{(n + 2) (n + 1) n!}{n!} \\ = \sum_{n = 0}^{\infty} {(- 1)}^{n} (n + 2) (n + 1) \end{matrix}$

The above series is divergent, thus the point -3 is not included in the convergent set. The convergent set for the given power series is. $x \in (- 3, - 1)$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In problems 1-6, determine the convergence set of the given power series.
$\sum_{n = 0}^{\infty} \frac{(n + 2)!}{n!} {(x + 2)}^{n}$

Step by Step Solution

Step 1:To Find the Radius of convergence

Step 2: Find the set of convergence

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Fundamentals Of Differential Equations And Boundary Value Problems

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Step by Step Solution

Step 1:To Find the Radius of convergence

Step 2: Find the set of convergence

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